text:
\section{Introduction}
Most living organisms exhibit daily biological rhythms. One of the most curious and difficult to understand is the sleep/wake cycle. Even though we are very familiar with sleep/wake, the reasons for sleep are not well understood. What is intriguing about the sleep-wake cycle is the complex interplay between the arousal and sleep states. Part of this is due to the cyclic alteration of the rapid eye movement (REM) and non-REM (NREM) stages of sleep.  The distribution of REM and NREM varies dramatically across the animal kingdom, to the point that a clear pattern has not emerged.  One consequence of this is that an animal model for human sleep has not been found \cite{Zimmer}. Another complication is that sleep/wake is sometimes thought to be synonymous with the 24 hour circadian clock.  This is not correct.  Although sleep/wake is coupled to the circadian clock, it is regulated by a neural system distinct from the circadian timing mechanism \cite{Semba}. Sleep/wake are also heavily influenced by two other factors.  One is \textit{homeostatic}, where the need for sleep increases with the length of the wake period.  The second is \textit{allostatic}, where the sleep/wake cycle is modified due to external behavioral events \cite{saper}. The circadian process is connected to the sleep-wake cycle, while a homeostatic process is inherent to the sleep-wake cycle. To be specific, it is now known that the sleep-wake cycle is independent of the diurnal stimulus, and is instead driven by homeostatic forces \cite{scammell,Jones,borbely}. Therefore, the inherent periodicity of the sleep/wake cycle is not due to a periodic forcing mechanism associated with the diurnal cycle but is an inherent characteristic of the  system.  The objective of this study is to derive, and then analyze, a model for the sleep/wake cycle that accounts for the circadian and homeostatic influences of sleep/wake, but not necessarily the allostatic affects.  This will include both REM and NREM sleep, as well as the waking state and observations about the associated brainwave activity of all states.

\begin{figure}[h]
	\begin{center}
		\includegraphics[width=5in]{Fig4.pdf}
	\caption{Approximate locations of areas pertaining to Sleep-Wake promoting neuronal groups.}
	\label{fig:fig4}
 \end{center}
\end{figure}

\subsection{Biological Background}
A basic assumption is that the brain has two states, sleep and wake, with sharp transitions between them.  There are multiple cell groups in the brain contributing to these states, but the monoaminergic nuclei (MN) have been identified as primarily responsible for promoting the awake state \cite{saper}.  Sleep is primarily promoted by the ventrolateral preoptic nucleus (VLPO), which is a group of cells associated with generating NREM and REM. A consequence of this distinct structural organization is that although the states are coupled, they have different regulatory mechanisms in the brain.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=4.5in]{Fig1.pdf}
\caption{The basic partitions of the states of sleep and wake for humans.}
\label{fig:fig1}
\end{center}
\end{figure}

\subsubsection{REM/NREM}

The REM and NREM phases of sleep have a well defined pattern (refer to Figure \ref{fig:fig1}).  Regardless of the conditions of sleep onset, a person always begins in an NREM period of the cycle.  After approximately 80-85 minutes the brain switches to the REM stage, and remains in this state for approximately 5-10 minutes before switching back to NREM.  This 90 minute pattern continues through the night with the REM-on intervals becoming progressing longer, and the NREM-on intervals shorter.  The last REM-on period is about 25 minutes.  There is no probability of waking naturally during an NREM interval, as the NREM component occurs during the periods of slow wave sleep. It is postulated that this is due to NREM being the most mentally restorative of the two processes, requiring little brain function and unconsciousness. Natural wake up occurs towards the end of the last NREM period. \cite{SRS} \\

There are three different ways of triggering sleep onset: exciting sleep active neurons, inhibiting wake active neurons or a combination of the two. The same idea applies to the triggering of wake onset.  Consequently, to investigate the source of sleep/wake brain function, it is necessary to examine the neurotransmitter activity of the system.

\subsubsection{Location Specific Neurotransmitter Activity}
The chemical activity of the brain involves the secretion of hormones and neurotransmitters from specific sections of the brain. The sleep-wake cycle is initiated and maintained by multiple distinct neuronal regions in the brainstem. These regions are located in the hypothalamus and are coupled to each other via electrical and neurochemical signaling. Figures \ref{fig:fig2} and \ref{fig:fig3} provide a schematic of location specific neurotransmitter signal flow in terms of excitatory and inhibitory mechanisms. Figure \ref{fig:fig4} shows the approximate locations of sleep and wake promoting areas in the human brain. Figure \ref{fig:fig5} provides a simple depiction of the switching between wake and sleep states and well as the switch between REM and NREM states, and the set of neurotransmitters associated with each, as well as their excitatory (+) or inhibitory (-) effects. The major neurotransmitter players are GABA (gamma-aminobutyric acid), orexin (hypocretin), histamine, acetylcholine, noradrenaline (norepinephrine), serotonin and dopamine. Each of these transmitters are located throughout the following neuronal groups: basal forebrain (BF), raphe nuclei (RN), laterodorsal tegmentum (LDT), pedunculopontine tegmentum (PPT), locus coeruleus (LC), lateral hypothalamus (LH), tuberomammillary nucleus (TMN), ventral periaqueductal gray region (vPAG), and ventrolateral preoptic nucleus (VLPO). Adenosine has been proposed to be the homeostatic accumulator of the need for sleep \cite{saper}. Acetylcholine effects muscle action, learning and memory. Norepinephrine effects one's control of alertness and wakefulness. Dopamine effects movement, focus and learning. Serotonin regulates mood, appetite, sleep, arousal and pain threshold. GABA contributes to sleep control and appetite control.  Also note that the sections of the brain we will be focusing on are the cerebral cortex, hypothalamus and thalamus, which play key roles in homeostatic maintenance. In this way they are the most relevant to the control of the human sleep/wake system. \\

Figure \ref{fig:fig6} shows one full period of the sleep-wake cycle model in terms of adenosine and GABA concentration levels. The point at which the curves cross indicate wake or sleep initializations, depending on whether one of the variables is increasing or decreasing. Wake Initialization occurs when $GABA$ is decreasing and $AD$ is increasing. Sleep initialization occurs when $AD$ is decreasing and $GABA$ is increasing.

\begin{figure}[h]
	\begin{center}
		\includegraphics[width=5in]{Fig2.pdf}
	\caption{Excitatory projections of neurotransmitter flow in the sleep/wake generating network.}
	\label{fig:fig2}
 \end{center}
\end{figure}

\begin{figure}[h!]
	\begin{center}
		\includegraphics[width=5in]{Fig3.pdf}
	\caption{Inhibitory projections of neurotransmitter flow in the sleep/wake generating network}
	\label{fig:fig3}
 \end{center}
\end{figure}

\begin{figure}[h]
	\begin{center}
		\includegraphics[width=5in]{Fig5.pdf}
	\caption{The cycle of NREM Sleep, REM Sleep and Wakefulness with the neurotransmitters characteristic of their stages.}
	\label{fig:fig5}
 \end{center}
\end{figure}

\begin{figure}[h]
	\begin{center}
		\includegraphics[width=5in]{Fig6.pdf}
	\caption{One full period of the sleep-wake cycle model.}
	\label{fig:fig6}
 \end{center}
\end{figure}

\newpage
\begin{sidewaystable}[h]
\centering
\vspace{0.25in}
\caption{Neurotransmitter Details}
\bigskip
\begin{tabular}{| p{0.7in} | p{0.34in}| p{0.7in} | p{1.15in} | p{1.15in} | p{0.8in} | p{0.7in} | p{0.7in} |}
\hline
Variable\footnote{Citations for the information found in this table are the following: \cite{adenosine1,Principles,scammell,adenosine_effects,chou_lu,arousal_systems,serotonin,Jones2,Dzirasa,Jones,Baumann,Arrigoni,adenosine2,Wu,Alreja,Khateb,Khateb2,John}} &  Region & Neurotrans -mitter & Receives Projections From: & Projects to: & Wake & NREM & REM\\
\hline
$GABA_{VLPO}$ & VLPO & GABA & LC, vPAG, RN, BF & LH, LC, LDT/PPT, vPAG, RN, BF & Inactive & High firing rate, high level. & Low firing rate, low level.  \\
\hline
$GABA_{BFs}$ & BF & GABA & BF, RN, LC, LH & BF, VLPO & Inactive & High firing rate, high level. & Inactive\\
\hline
$GABA_{BFw}$ & BF & GABA & BF, RN, LC, LH & BF, VLPO & Active & Inactive & Active \\
\hline
$OX$ & LH & Orexin & LH, VLPO, RN, TMN & LH, TMN, RN, BF & High firing rate, high level. & Low firing rate, low level. & Inactive \\
\hline
$H$ & TMN & Histamine & TMN, LH, VLPO & TMN, LH & Tonic, low firing rate, high level. & Low firing rate, low level. & Inactive \\
\hline
$ACh_{BF}$ & BF & Acetylcholine & BF, RN, LC, LH & BF, VLPO & High firing rate, high level. & Low firing rate, low level. & High firing rate, high level. \\
\hline
$ACh_{LDT/PPT}$ & LDT/ PPT & Acetylcholine & BF, VLPO & VLPO & Fast cortical rhythms & Inactive & Fast cortical rhythms. \\
\hline
 $NA$ & LC & Noradrenaline & LC, VLPO & LC, BF, RN, VLPO & Tonic, low firing rate, high level. & Low firing rate, low level. & Inactive \\
  \hline
 $S$ & RN & Serotonin & RN, LH, LC, VLPO & RN, VLPO, LH, BF & Tonic, low firing rate, high level. & Low firing rate, low level. & Inactive \\
\hline
$DA$ & vPAG & Dopamine & vPAG, VLPO & vPAG, VLPO & No rate change. Elevated levels. & No rate change. & No rate change. \\
\hline
$AD$ & All Cells & Adenosine & Not Applicable & Not Applicable & Levels increase with prolonged wake in the BF. & Levels fall during recovery sleep. & Levels fall during recovery sleep. \\
\hline
\end{tabular}
\end{sidewaystable}

\begin{figure}[t]
	\begin{center}
		\includegraphics[width=4in]{Fig7.pdf}
	\caption{Components of the human sleep/wake cycle as well as their associate neurotransmitters}
	\label{fig:fig7}
 \end{center}
\end{figure}

Adenosine, which is a neuromodulator, plays a major role in the sleep-wake process. It is a byproduct of adenosine-triphosphate (ATP), the main ``energy currency" of the brain, whenever neurotransmitters are fired, adenosine is produced. Adenosine is also continuously converted back to ATP via the metabolic processes that use it as an energy source \cite{metabo}. The adenosine concentration that we are most interested in is that of the basal forebrain. While inhibiting the expression of all wake active neurotransmitters and disinhibiting the expression of those found in the VLPO \cite{adenosine_effects}, adenosine levels in the basal forebrain increase during prolonged wake and fall during recovery sleep, providing an intrinsic periodic force associated with waking \cite{adenosine1,adenosine2}. The linear relationship between between adenosine accumulation and sleep propensity provides support for using adenosine as representative of sleep propensity. GABA in the VLPO can be viewed as the other major player in the sleep-wake system. $GABA_{A}$ receptors in the brain mediate the \textit{tonic} (or slow) conductance \cite{Winsky}. This slower type of signaling plays a key role in cell excitability and sleep homeostasis \cite{Winsky}. Neuron populations in the VLPO are inactive during the vigilance state, and have a high firing rate and level during NREM, especially during delta wave (slow wave) sleep \cite{Winsky}.  Being the major inhibitory neurotransmitter in the central nervous system, GABA is a key player in many physical systems.

Referring to the Table 1, we can see the detailed activity associated with each neurotransmitter and brain area, allowing for the development of a system of variables. As a simple example, $OX_{LH}$ is the variable associated with the chemical concentration of orexin in the lateral hypothalamus. Since we only discuss orexin in one location, the subscript LH can be dropped. Orexin expression is promoted by acetylcholine in the LDT/PPT and noradrenaline in the LC. Orexin expression is inhibited by adenosine, serotonin, GABA and noradrenaline. Orexin promotes the release of noradrenaline, serotonin, histamine, GABA in the basal forebrain, acetylcholine, and dopamine, as well as promoting its own release (autoexcitation). These dynamics indicate the need for a linear combination of acetylcholine, noradrenaline, serotonin, dopamine, histamine, GABA, adenosine, and orexin to express the rate of change of OX concentration.\\

\section{Materials and Methods}
In order to test the functionality of the model, we use numerical methods to determine how the system responds to a variety of perturbations.  In the calculations to follow, MATLAB 2009a was used to run all the programs written for the Human Sleep/Wake system described here. However, a fourth order Runge-Kutta method was used for solving the system of equations, instead of using one of the MATLAB ode solvers because of the unique perturbations used in the sleep experiments described.

\section{Theory and Calculation}

\subsection{Previous Work}
A good survey of the early sleep/wake models can be found in \cite{strogatz2}.  We will only comment here on those most relevant to our development. \\

In a recent paper by Phillips and Robinson \cite{Phillips}, a model was proposed in order to improve quantitative understanding of the human sleep/wake cycle, which is similar to the intentions of this study. They use the following three neuronal populations to model the human ascending arousal system: monoaminergic (MA) nuclei, acetylcholine related (ACh) nuclei, and populations in the VLPO. The VLPO is then split into circadian and homeostatic drives, which are represented by periodic forcing functions. Each population has its dynamics modeled by a sum of mean cell body potentials and mean neuronal firing rates subjected to the periodic forcing of the VLPO. While their model consistently predicts sleep/wake cycle dynamics observed in previous studies, it is important to note that the homeostatic and circadian drives are represented as periodic forcing rather than dynamics inherent to the physical system. \\

\cite{Cecilia} developed a model of the network of neurons controlling behavioral state transitions in mice, while also identifying special features of human and mouse sleep. It is similar to the Phillips and Robinson model (as well as the model being developed here) in that it defines three main processes being modeled: Sleep activity, Wake activity, and REM activity. Each population is defined by a set of three equations, determined by a wide range in timescales. It incorporates homeostatic REM and NREM sleep drive and uses known sleep active neurotransmitters to determine parameters. It then utilizes extensive experimental data gathered from mouse sleep/wake states to verify the accuracy of the model. A limitation of their approach concerns the separation of vigilance states, which are characterized in a non-quantitative fashion. Similarly, ``activity states" are used rather than specific neurotransmitter concentration and/or neuron firing data. \\

Taking a somewhat different viewpoint,  \cite{best} use the ``flip-flop" dynamics proposed in \cite{saper}  to construct a biologically-based mathematical model that accounts for several features of the human sleep/wake cycle.  This differs from our model in several ways, but two examples are that they assume, incorrectly, that the SCN projects directly to the VLPO, and an external periodic forcing mechanism is used to drive the model. \\

In another recent paper by \cite{DinizBehn2010}, a model of the rat sleep/wake regulatory network is formed utilizing similar techniques as those used to model a mouse sleep/wake regulatory network in her aforementioned work \cite{Cecilia}. In this model,  neurotransmitter concentrations  are coupled  to neuronal group firing rates to create a more complete view of the hypothalamic sleep-wake regulatory network. The model is composed of three parts: wake/sleep active neuron population firing rates, wake/sleep active neurotransmitter concentrations and a homeostatic regulator. All physical parameters were determined using extensive experimental data. This paper emphasizes the importance of utilizing neuron firing rates as well as neurochemical concentrations to form a physiologically accurate model of sleep/wake regulation. A downside of this model lies in the use of steady state saturation functions and Heaviside functions. Rather than using inherent neurotransmitter dynamics, the functions of neurotransmitter release are approximated using sigmoidal functions. The homeostatic drive is a physically unquantifiable force composed of Heaviside switching mechanisms. \\

\subsection{Model Formulation}

In this model for the sleep/wake cycle, we are considering three principal states.  They are the wake state ($W$), the NREM sleep state ($S$) and the REM sleep state ($R$). Table 1 provides a framework for the detailed interactions of all the neurotransmitters involved in the sleep-wake system.  Given the number of variables involved,  before deriving the equations it is worth describing the dynamics in more qualitative terms.  \\

The sleep-specific and wake specific neurotransmitters operate on a \textit{slow time scale}, measured in terms of hours, while the REM and NREM specific neurotransmitters operate on a \textit{fast time scale system}, measured in terms of minutes. The two neurochemicals associated with the slow time scale are adenosine in the BF and GABA in the VLPO. We use these two concentrations as variables indicating promotion of wake and sleep. High adenosine concentrations in the BF promote wakefulness, while high GABA concentrations and fast firing rates in the VLPO promote sleep.  Adenosine's slow build up and decay is associated with wakefulness, while GABA in the VLPO is associated with sleep. These observations are the basis for the hypothesis that adenosine is the principal component in determining wake, and $GABA_{VLPO}$ is the principal component in promoting sleep \cite{Winsky,adenosine1,adenosine2}. We also know from these references that AD disinhibits the VLPO during slow wave sleep, providing a direct connection between the building adenosine sleep pressure during waking and the gradual relief of this pressure during sleep. The other neurotransmitters are assumed to be associated with the fast time scale dynamics.  Adenosine and $GABA_{VLPO}$ fluctuate with a period of 24 hours, while the other neurotransmitters fluctuate with a period of several minutes. \\

An important inherent quality of the sleep-wake cycle is that it should maintain it's state transitions in the absence of external periodic forcing. Mathematically, this indicates the need for the system to have a \textit{stable (attracting) limit cycle}.   This solution should also be stable to certain perturbations, such as those that might occur from genetic mutations causing the production of too much or too little of a neurotransmitter, or lacking the proper neuron groups to produce a neurotransmitter at all. Perturbations may also occur via the removal of specific neuron groups, blockage of neurotransmitter bonding sites, and external forcing. It is well known from experimental data that perturbations to neurotransmitters don't alter the two-state nature of the sleep-wake system \cite{adenosine1,adenosine2,Arrigoni,OXKO,John,Lee,Khateb,Khateb2,arousal_systems,Dzirasa,Lu,Crochet,Gervasoni,Baumann}. As an example, cholinergic agonists applied to the LC in cats leads to increased REM sleep, but the sleep-wake cycle as a whole remains periodic \cite{Crochet}. Therefore one of the central goals will be to locate the area in the phase plane of $GABA_{VLPO}$ and $AD$ where a limit cycle exists and is stable.  \\

\subsubsection{Kinetics}
For each neurotransmitter involved in the sleep-wake system, we identify a location specific variable representing the effective neurotransmitter concentration. We assume first order kinetics for the neurotransmitters operating on the fast time scale. The fast time system may be formed from the observations compiled about the inhibitory and excitatory nature of the neurotransmitters in Table 1 combined with the dynamics arising in a synapse.  If $X_{i}$ is representative of a neurotransmitter in region $i$, we have
\begin{equation}
\frac{dX_{i}}{dt} = {\rm Release(t)} - {\rm Uptake(t)} - {\rm Degradation(t)}. \label{eq:eq1}
\end{equation}
Note that inhibitory effects are considered pre-synaptic and are therefore controlling flow from the neuron into the synaptic gap. Despite the fact that inhibitory and excitatory effects can occur pre- or post-synaptically, for simplicity we will assume the blocking or binding process of inhibition as only controlling flow into the gap. The Release function contains rate parameters and neurotransmitter variables that excite and inhibit the expression of $X_{i}$. The Uptake function depends on $X_{i}$ and its associated vesicular uptake transporter, if existent. The Degradation function depends on $X_{i}$ and an associated rate of decay. The uptake transporters are modeled using known logistic dynamics from neurotransmitter uptake transporter microdialysis measurements \cite{neurotransport}. \\

As an example, the release of histamine is promoted by orexin, noradrenaline and acetylcholine in the BF and inhibited by GABA in the VLPO. Histamine is autoinhibitory, like most of the monoamines. The equation for histamine's rate of change, like all of the other neurochemicals involved, also requires an adenosine term. Histamine in the human sleep-wake system has no known uptake mechanism, and so only degrades as a method of removal. Therefore the equation for histamine is
\begin{equation}
 \frac{dH}{dt} = c_{i}OX + c_{i+1}NA + c_{i+2}ACh_{BF} + c_{i+3}AD - c_{i+4}GABA_{VLPO} - (c_{i+5} - c_{i+6})H \notag
\end{equation}
where the $c_{i}$'s are rate coefficients. The rest of the equations are formulated in the same fashion.

\bigskip\noindent
\textbf{Fast Time System}
\begin{align}
\frac{dGABA_{BFw}}{dt} &= c_{1}ACh_{BF} + c_{2}OX + c_{3}AD - c_{4}GABA_{BFw} - \notag \\
& \qquad c_{5}(GA1)(GABA_{BFw} ) \label{eq:eq2}\\
\frac{dGABA_{BFs}}{dt} &=  c_{6}AD + c_{7}NA - c_{8}GABA_{BFs} - c_{9}(GA1)(GABA_{BFs}) \label{eq:eq3}\\
\frac{dOX}{dt} &= c_{10}ACh_{LDT/PPT} + (c_{11} - c_{12})NA + (c_{13}-c_{14})OX + c_{15}AD - \notag \\
& \qquad c_{16}S - c_{17}GABA_{VLPO} + c_{18}GABA_{BFs} \label{eq:eq4}\\
\frac{dH}{dt} &= c_{19}OX + c_{20}NA + c_{21}ACh_{BF} + c_{22}AD - c_{23}GABA_{VLPO} - \notag \\
& \qquad (c_{24} - c_{25})H \label{eq:eq5} \\
\frac{dACh_{BF}}{dt} &= c_{26}OX + c_{27}H + c_{28}AD - c_{29}S + c_{30}GABA_{BFs} - c_{31}ACh_{BF} \label{eq:eq6} \\
\frac{dACh_{LDT/PPT}}{dt} &= c_{32}OX + c_{33}H + c_{34}AD + c_{35}NA - c_{36}S - c_{37}ACh_{LDT/PPT} \label{eq:eq7} \\
\frac{dNA}{dt} &= c_{38}OX + c_{39}ACh_{BF} + c_{40}AD - c_{41}GABA_{VLPO} - \notag \\
& \qquad (c_{42} - c_{43})NA - c_{44}(hNET)(NA) \label{eq:eq8} \\
\frac{dS}{dt} &= c_{45}OX + c_{46}H + c_{47}NA + c_{48}AD - c_{49}GABA_{VLPO} - \notag \\
& \qquad (c_{50}  - c_{51})S - c_{52}(hSERT)(S) \label{eq:eq9} \\
\frac{dDA}{dt} &= c_{53}OX + c_{54}S + c_{55}ACh_{LDT/PPT} + c_{56}NA + c_{57}AD - \notag \\
& \qquad c_{58}GABA_{VLPO} - c_{59}(hDAT)(DA) - c_{60}DA  \label{eq:eq10}
\end{align}

The uptake transporter values are contained in the uptake terms in equations (\ref{eq:eq2}), (\ref{eq:eq3}), (\ref{eq:eq8}), (\ref{eq:eq9}) and (\ref{eq:eq10}). Their values are found from microdialysis experiments with human cerebrospinal fluid (CSF), and are as follows \cite{neurotransport}: $GA1 = 1$,
$hNET = 0.457$,
$hSERT = 0.463$,
$hDAT = 1.22 $.\\

\subsubsection{Uptake Mechanisms}

The neurotransmitters involved in the sleep-wake system, as with any neuronal network, have natural rates of removal from the synaptic gap once released. These rates can either be contributed to by degradation, neurotransmitter uptake or a combination of both. All of the neurotransmitters involved degrade over time, but only some have neurotransmitter uptake mechanisms. We will not be considering any of the smaller scale chemical dynamics of the neurotransmitter uptake mechanisms. We care only whether a neurotransmitter is removed  by an uptake mechanism, and will not differentiate between plasma membrane-bound transportation or receptor uptake. The key take away from the uptake transportation data is that only GABA, noradrenaline, serotonin and dopamine have significant synaptic uptake transport mechanisms in humans. These mechanisms are characterized by the data given constants $GA1$, $hNET$, $hSERT$ and $hDAT$ for GABA, noradrenaline, serotonin and dopamine, respectively \cite{neurotransport}. These neurotransmitter transporters appear in the uptake term of the equations, always multiplied by the neurotransmitter it is dependent on to indicate a linear removal rate. As an example, $hSERT$ appears in equation 9 multiplied by $-c_{52}S$, so as to act as a removal mechanism for serotonin when it is present. However, $hSERT$ does not multiply the natural degradation term $-c_{50}S$.\\

\subsubsection{Nonlinear Dynamics}
There are two neurotransmitters, $AD$ and $GABA_{VLPO}$, operating on a slow time scale that define the sleep-wake cycle. The qualitative portrait in phase space is known from basic sleep-wake cycle activity. We can construct the desired closed orbit and concentric curves from the nullclines of the $AD$ and $GABA_{VLPO}$ equations. We identify the bounding region in the phase space activity by the box contained between $(AD, GABA_{VLPO})  = (AD_{max},GABA_{min})$ and $(AD_{min},GABA_{max})$. We choose small values of $AD_{min} = GABA_{min} = 0.001$ rather than exactly zero to prevent inclusion of a potentially unstable steady state. We can scale the size of this box according to the values of $AD_{max}$ and $GABA_{min}$. These two points are represented by the intersections of the $AD$ and $GABA_{VLPO}$ nullclines. To describe the activity between these two points, we begin with the fact that when $AD$ is at it's maximum value, $GABA_{VLPO}$ is at it's minimum value (0.001). This signifies that at the peak of wake activity, sleep activity is at its minimum. As we move along the $AD$ nullcline, the concentration of AD decreases to 0.001 as the concentration of $GABA_{VLPO}$ increases to $GABA_{max}$. From the point $(GABA_{VLPO}, AD) = (GABA_{max},0.001)$, we move along the $GABA_{VLPO}$ nullcline slowly until we reach the point $(GABA_{VLPO}, AD) = (0.001, AD_{max})$ again, creating a closed orbit. Figure \ref{fig:fig8} shows this movement in the phase plane, as well as showing the location of the fixed point and neighboring trajectories. Everything is in non-dimensional form, and therefore should be thought of in terms of threshold levels rather than micro-moles per liter. \\
\begin{figure}[t]
	\begin{center}
		\includegraphics[width=5in]{Fig8.pdf}
	\caption{The ideal activity of the two neurotransmitter nullclines in the $GABA_{VLPO}-AD$ phase plane.}
	\label{fig:fig8}
 \end{center}
\end{figure}

The system is made up of two widely separated time scales: the slow, 24 hour time scale that is governed by the build-up and decay of adenosine, and the fast, O(1) (minute) time scale that involves interactions of all the sleep and wake active neutrotransmitters, but is regulated by the oscillation of the slow time scale. \\

We begin by forming the slow time system that will later be shown to be responsible for the limit cycle dynamics of the system. This system is comprised of two equations: the adenosine equation that fluctuates with wake activity, and the GABA in the VLPO equation that fluctuates with NREM sleep activity. We consider NREM sleep to be the ground state of the sleep-wake system, and all other oscillations to be deviations thereof. The adenosine equation must involve a constant source term, due to adenosine's constant creation as a by-product of neuronal activity, as well as a decay term that is dependent on adenosine's natural rate of degradation as well as it's loss from the synaptic gap through the its disinhibition of GABA in the VLPO \cite{adenosine_effects}. It is well known that mechanism for receptor binding is nonlinear \cite{neurotransport}, and so the disinhibition effect of $GABA_{VLPO}$ via adenosine should be nonlinear. It also follows that this nonlinear term should appear in both the equations for $AD$ and $GABA_{VLPO}$ because adenosine's disinhibition of the VLPO both depletes the concentration of adenosine while simultaneously (and with the same rate) adding to the $GABA_{VLPO}$ concentration. The $GABA_{VLPO}$ equation must contain the uptake term for GABA, a degradation term and a contribution from adenosine as well as the inhibitory and excitatory of other neurotransmitters. A constant loss term is required in the $GABA_{VLPO}$ equation as a representation of its quick breakdown time in the synaptic gap \cite{Winsky}. Though it is being broken down quickly and continuously, $GABA_{VLPO}$ still provides a tonic effect on the sleep-wake system as a whole \cite{Winsky}. Based on these requirements as well as the observations of the glycolysis reaction \cite{Selkov}, we obtain the following slow time system: \\

\noindent \textbf{Slow Time System}
\begin{equation}
\frac{dAD}{dt} = k_{1} - k_{2}AD - (GABA_{VLPO})^{2}(AD)
\label{eq:eq11}
\end{equation}

\begin{equation}
\frac{dGABA_{VLPO}}{dt} = -\epsilon - k_{3}GABA_{VLPO} + k_{4}AD + (GABA_{VLPO})^{2}(AD) \label{eq:eq12}
\end{equation}

\bigskip\noindent
where
$$\epsilon = \mu - (a_{1}ACh_{BF} + a_{2}ACh_{LDT/PPT} + a_{3}NA + a_{4}S + a_{5}DA)$$
To provide a more physical explanation of the assumed reaction system, we know that adenosine is a bi-product of the constant source of energy in the nervous system, adenosine triphosphate (ATP). We know that adenosine is continuously converted back into ATP \cite{metabo}.  From experiments in the VLPO, we know that activation of VLPO neurons through adenosine-mediated inhibition contributes to the somnegenic actions of adenosine, and only adenosine acts directly on the VLPO in the sleep-wake system \cite{adenosine_effects,Arrigoni,adenosine1,adenosine2}. During wake, increased levels of adenosine promote sleep by disinhibiting the release of GABA in VLPO. Adenosine levels in this area increase during wake and decrease during sleep \cite{adenosine_effects}. In this way the physical foundation for the glycolysis system is similar to and provides a good framework for our slow time scale system. \\

\subsubsection{Characterizing REM/NREM}

REM and NREM sleep are determined by a nonlinear combination of neurotransmitters.  Specifically, $GABA_{VLPO}$ promotes the state of REM sleep, as does acetylcholine from both the LDT/PPT and the BF. While NREM can be seen as an absence of the vigilance state and the contributions from GABA, REM sleep must incorporate contributions from neurotransmitters active in the wake state as well as those in the NREM state. Letting $R(t)$ represent the REM sleep state, we assume that $R'' + \Phi R'+ \Gamma R= 0$. To determine  $\Phi$, note that the transition from the waking state to the sleeping state, and vice versa, occurs when $GABA_{VLPO} = AD$.  Given that the REM state has an asymptotically stable steady state during wake, which occurs when  $AD>GABA_{VLPO}$, and has a limit cycle during sleep, which occurs when $AD<GABA_{VLPO}$, then $\Phi$  contains a term of the form $(GABA_{VLPO} - AD)(R^2-R_0^2)$.  When entering wake, so $GABA_{VLPO}=AD$ and $\dot{AD}>0$, there is an impetus for the impending dormancy of REM and the resulting mathematical requirement is that $\Phi(\dot{AD},AD)>0$. Likewise, when entering sleep, so $GABA_{VLPO}=AD$ and $\dot{AD}<0$, the requirement for the oncoming REM limit cycle is $\Phi(\dot{AD},AD)<0$. This gives us

\begin{equation}
\frac{d^2 R}{dt^2} + \left [\alpha (GABA_{VLPO} - AD)(R^2 - R_0^2) + \beta \frac{dAD}{dt} \right ] \! \frac{dR}{dt} + \Gamma R = 0 \,. \label{eq:eq14}
\end{equation}

\subsubsection{Some Special Cases}
Given the complexity of (\ref{eq:eq14}), it is worth considering the solution in particular situations and then using this to help estimate the oscillatory response of equation (\ref{eq:eq14}).  For example,  the period for $R$ can be determined from the fact that there  are  three to four normal REM oscillations  over a normal night's sleep.   So, when $GABA_{VLPO} = AD$,

\begin{equation}
\frac{d^2 R}{dt^2} + \beta \frac{dAD}{dt}\frac{dR}{dt} + \Gamma R = 0 \,. \label{eq:eq15}
\end{equation}

\noindent
Given the slowly varying nature of $AD$, then $R = e^{rt}$, where $r =\frac{1}{2}( -\beta u \pm \sqrt{\beta^2 u^2 - 4\Gamma})$ and $u=(AD)'$.  Since $u$ strongly controls the decay, and to have $\Gamma$ control the number of REM oscillations, we require $r = -u \pm i \sqrt{\gamma}$, where $\sqrt{\gamma}$ is approximately the number of oscillations. Thus, $\beta = 2$ and $\Gamma = \gamma + u^{2}$.  From this we have that
\begin{equation}
\frac{d^2 R}{dt^2} + 2 \! \left [(GABA_{VLPO} - AD)(R^2 - 1.3) + \alpha \frac{dAD}{dt} \right ] \!  \frac{dR}{dt} +   \! \left (\gamma + \left (\frac{dAD}{dt}\right)^2 \right )  \! R= 0 \, . \label{eq:eq16}
\end{equation}

\noindent
where $\alpha $ is determined numerically and $8< \gamma<16$. \\

Based on Lienard's Theorem, we know that the above oscillator has a unique, stable limit cycle \cite{strogatz1}. Thus the REM oscillations will continue to function in a stable, periodic fashion even when subjected to reasonable perturbations. Evidence of this result is in the Orexin Knockout section. \\

\subsection{Analysis}
To find the values of the coefficients of the system and to ensure stability, we present several methods of analyzing the  equations. We begin with the limit cycle coming from the Slow Time System.   As stated earlier, the adenosine equation drives the wake state dynamics, and the GABA equation drives the sleep state dynamics.  The specific equations are

\begin{equation}
\frac{d}{dt} AD = k_{1} - k_{2}AD - (GABA_{VLPO})^{2}(AD) \label{eq:eq17}
\end{equation}
\begin{equation}
\frac{d}{dt} GABA_{VLPO} = -\epsilon - k_{3}GABA_{VLPO} + k_{4}AD + (GABA_{VLPO})^{2}(AD) ,
\label{eq:eq18}
\end{equation}

\noindent
where
$$\epsilon = \mu - (a_{1}ACh_{BF} + a_{2}ACh_{LDT/PPT} + a_{3}NA + a_{4}S + a_{5}DA) .$$

It is not possible to determine the rate constants with the available experimental evidence.  So, we take a different tack and establish bounds on the constants that will produce the responses known to occur in sleep and wake.  To begin, we compute the nullclines of the slow time system in order to create a trapping region for the limit cycle and to determine the coefficients of the system by using an approximate fixed point. We have four coefficients to determine ($k_1$, $k_2$, $k_3$, $k_4$) as well as reasonable values for the parameter $\epsilon$. Plugging the approximate fixed point into both nullclines will give us two equations with five unknowns. In order to make the system well determined, we observe the maxima and minima of the equations as well as the linearized slow time system with the approximate fixed point. Knowing the form of the determinant and the trace of this system needed to maintain stability, we can determine the rest of the unknowns. \\

Now, solving the nullcline equations obtained from \ref{eq:eq17} and (\ref{eq:eq18}) for $AD$ and we have
$$AD = \frac{k_{1}}{k_{2}+GABA_{VLPO}^2} \, ,$$
and
$$AD = \frac{k_{3}GABA_{VLPO} - \epsilon}{k_{4}+(GABA_{VLPO})^2} \, .$$

\noindent
Setting these equal to each other, we have three points of intersection, only one of which is real.  Using $GABA_{VLPO} \in [0,2]$ and $AD \in [0,2]$ as the boundaries of  the trapping region, we can approximate a fixed point inside the closed orbit at $GABA_{VLPO} = 0.703$ and $AD = 0.823$.  This provides us with two of the needed equations.\\

We also know that each of these equations is zero separately when $AD = AD_{max}$ and $GABA_{VLPO} = GABA_{min}$, as well as when $AD = AD_{min}$ and $GABA_{VLPO} = GABA_{max}$. We let $AD_{max} = GABA_{max} = 2$ and $AD_{min} = GABA_{min} = 0.01$.  Taking $\epsilon = 0.3$, and using the stated values for the max and min concentrations it follows that
$k_{1} = 0.49$,
$k_{2} = 0.1$,
$k_{3} = 0.3$.
$k_{4} = 0.15$.
 We also note that $0.29 < \epsilon \leq 0.32 $ for the system to remain stable. \\

\subsubsection{Remaining Coefficient Estimation}

Now that the coefficients for the slow time system are determined, we can use the requirements that the eigenvalues have either pure imaginary form or negative real parts in the complex form to determine the coefficients of the entire system. We have the solution vector of the linearized system as $\mathbf{x} =
[ GABA_{BFw},
GABA_{BFs},
OX,
H,
ACh_{BF} ,$
$ACh_{LDT/PPT} ,
NA ,
S ,
DA ,
AD ,
GABA_{VLPO}
]^T$. The matrix of coefficients and linearized values for the system are  given in Table \ref{tb:1.euler}.

\begin{sidewaystable}
\centering
\vspace{1in}
\renewcommand\arraystretch{1.5}
\renewcommand\tabcolsep{2pt}
B =
\begin{tabular}{p{0.5in} p{0.5in} p{0.5in} p{0.5in} p{0.5in} p{0.5in} p{0.7in} p{0.7in} p{0.6in} p{0.6in} p{1in}}
\footnotesize{-$c_{1}$-GA1} & 0 & \footnotesize{$c_{2}$} & 0 & \footnotesize{$c_{3}$} & 0 & 0 & 0 & 0 & \footnotesize{$c_{4}$} & 0 \\
0 & \footnotesize{-$c_{5}$-GA1} & 0 & 0 & 0 & 0 & -$c_{6}$ & 0 & 0 & -$c_{7}$ & $c_{8}$ \\
0 & -$c_{9}$ & \footnotesize{$c_{10}$-$c_{11}$} & 0 & 0 & $c_{12}$ & \footnotesize{$c_{13}$-$c_{14}$} & -$c_{15}$ & 0 &  $c_{16}$ & $-c_{17}$ \\
0 & 0 & $c_{18}$ & \footnotesize{$c_{19}$-$c_{20}$} & $c_{21}$ & 0 & $c_{22}$ & 0 & 0 & $c_{23}$ & -$c_{24}$ \\
0 & -$c_{25}$ & $c_{26}$ & $c_{27}$ & -$c_{28}$ & 0 & 0 & -$c_{29}$ & 0 & $c_{30}$ & 0 \\
0 & 0 & $c_{31}$ & $c_{32}$ & 0 & -$c_{33}$ & -$c_{34}$ & -$c_{35}$ & 0 & $c_{36}$ & 0 \\
0 & 0 & $c_{37}$ & 0 & $c_{38}$ & 0 & \footnotesize{-$c_{39}$-$c_{40}$-hN} & 0 & 0 & $c_{41}$ & -$c_{42}$ \\
0 & 0 & $c_{43}$& $c_{44}$ & 0 & 0 & $c_{45}$ & \footnotesize{-$c_{46}$-$c_{47}$-hS} & 0 & $c_{48}$ & -$c_{49}$ \\
0 & 0 & $c_{50}$ & 0 & 0 & $c_{51}$ & -$c_{52}$ & $c_{53}$ & \footnotesize{-$c_{54}$-$hD$} & $c_{55}$ & -$c_{56}$ \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \footnotesize{-$0.1 - G_{V}^2$} & \footnotesize{-$2G_{V}A$}\\
0 & 0 & 0 & 0 & \footnotesize{-$c_{57}$} & \footnotesize{-$c_{58}$} & \footnotesize{-$c_{59}$} & \footnotesize{-$c_{60}$} & \footnotesize{-$c_{61}$} &  \footnotesize{ $0.15 + G_{V}^2$} & \footnotesize{-$0.3+2G_{V}A$}
\end{tabular}
\caption{Matrix B of the linearized system. Note that  $hN=hNET$, $hS=hSERT$, $hD=hDAT$, $G_{V} = GABA_{VLPO}$ and $A = AD$.}
\label{tb:1.euler}
\end{sidewaystable}

In order for the system to be stable, we require that the real parts of the eigenvalues to be negative. Restricting the coefficients to all be between 0 and 1, we use an algorithm that generates coefficients that keep the real parts of the eigenvalues negative. We also note that the coefficients in $GABA_{VLPO}$ equation need to be small compared to the rest of the coefficients in order to keep $\epsilon$ bounded.

\begin{table}
\begin{tabular}{p{1in} p{1.2in} p{1in} p{1in}}
\hline
Parameter & Value & Parameter & Value\\
\hline
$GA1$ & 1 & $c_{27}$ & 0.10973\\
$hNET$ & 0.457 & $c_{28}$ & 0.32943\\
$hSERT$ & 0.463 & $c_{29}$ & 0.57879\\
$hDAT$ & 1.22 & $c_{30}$ & 1\\
$k_{1}$ & 0.49 & $c_{31}$ & 0.02091\\
$k_{2}$ & 0.1 & $c_{32}$ & 0.12648\\
$k_{3}$ & 0.3 & $c_{33}$ & 0.23472\\
$k_{4}$ & 0.15 & $c_{34}$ & 0.57122\\
$\mu$ & 0.3 & $c_{35}$ & 0.02332\\
$\alpha$ & 1 & $c_{36}$ & 1\\
$\gamma$ & 8 & $c_{37}$ & 0.61305\\
$c_{1}$ & 0.75709 & $c_{38}$ & 0.06864\\
$c_{2}$ & 0.28014 & $c_{39}$ & 0.08638\\
$c_{3}$ & 0.61048 & $c_{40}$ & 0.1\\
$c_{4}$ & 0.76636 & $c_{41}$ & 0.543\\
$c_{5}$ & 0.32431 & $c_{42}$ & 0\\
$c_{6}$ & 0.83153 & $c_{43}$ & 0.13822\\
$c_{7}$ & 0.03471 & $c_{44}$ & 0.35956\\
$c_{8}$ & 0.01 & $c_{45}$ & 0.11839\\
$c_{9}$ & 0.19577 & $c_{46}$ & 0.13753\\
$c_{10}$ & 0.79157 & $c_{47}$ &0.1\\
$c_{11}$ & 0.97026 &  $c_{48}$ & 0.537\\
$c_{12}$ &-1 &  $c_{49}$ & 0\\
$c_{13}$ & 0 & $c_{50}$ & 0.67749\\
$c_{14}$ & 0.36341 & $c_{51}$ & 0.53609\\
$c_{15}$ & 0.70633 & $c_{52}$ & 0.36464\\
$c_{16}$ & 0.1 & $c_{53}$ & 0.59591\\
$c_{17}$ & 0.97643 & $c_{54}$ & 0.75091\\
$c_{18}$ & 0.56740 & $c_{55}$ & 0.1\\
$c_{19}$ & 0.91859 & $c_{56}$ & 0.22\\
$c_{20}$ & 0.50364 & $c_{57}$ &$10^{-5}$\\
$c_{21}$ & 0.23758 & $c_{58}$ & $10^{-5}$\\
$c_{22}$ & 0.1 & $c_{59}$ & $10^{-5}$\\
$c_{23}$ & 1 & $c_{60}$ &$10^{-5}$\\
$c_{24}$ & 0 & $c_{61}$ &$10^{-5}$ \\
$c_{25}$ & 0.92037 & & \\
$c_{26}$ & 0.04185 & & \\

\hline
\end{tabular}
\caption{Coefficient and parameter values used in the sleep-wake cycle model.}
\end{table}

\subsubsection{Sub-Systems}
When analyzing such a large system of equations with undetermined coefficients, it is helpful to break the large system into smaller sub-systems and observe the dynamics on a local scale. We can utilize the neurophysiology of the physical neurotransmitter sub-systems to determine the coefficients as well as the stability profiles of the individual sub-systems. We break the system up into to following categories:
NREM Specific,
Sleep Specific (NREM Inclusive),
Wake Specific,
REM + NREM Specific,
REM + Wake Specific,
Wake Inclusive,
Sleep Inclusive,
REM Inclusive.
Note that there is no set of equations associated with REM only. The categories represent what neuron populations are active during the corresponding states. We observe the steady-state behavior of each sub-system. Since we know the relative magnitude of the neurotransmitter amounts as well as their dynamic behavior during the states of REM, NREM and Wake, the coefficients of these smaller, more manageable systems should be more easily attainable. This also allows us to observe the more manageable systems for potentially extreme behavior, and provides us with bounds to prevent this from happening. This analysis does not give us specific values for the system coefficients, knowing the relative values makes visualizing the system easier as well as simplifying the eigenvalue analysis. \\

\subsubsection*{NREM Specific}
The NREM specific system involves the neuron population only active during the state of NREM sleep. The neuron population that is only active in NREM sleep is the GABAergic population in the BF. We assume all equations but $\frac{dGABA_{BFs}}{dt}$ to be equal to zero, making each neurotransmitter other than $GABA_{BFs}$ constant, and assume the other neurotransmitters in the equation to be at their steady state values when only $GABA_{BFs}$ is active. This is not necessarily a physiologically accurate view of the system for an extended length of time, but we can view a snapshot of this activity. We follow this same criteria for the rest of the small systems.

\subsubsection*{Wake Specific}
The Wake specific system involves neuron populations with the most activity during the wake state. The monoaminergic neuron populations (including OX, H, NA, S and DA) are most active in this state.

\subsubsection*{Sleep Specific (NREM Inclusive)}
The Sleep Specific system involves neurotransmitters that promote sleep states (REMS and NREMS). The populations that are active in this state are GABAergic populations in the VLPO and BF.

\subsubsection*{REM + Wake Specific}
The REM and Wake Specific system involves neurotransmitters that promote REMS and the Wake state. The neuron populations that are active during these states are GABAergic populations in the BF and cholinergic populations in the BF and LDT/PPT.

\subsubsection*{REM + NREM Specific}
The REM and NREM Specific system involves neurotransmitters promoting REM sleep and the NREM sleep. The neuron population that is active during these states is the GABAergic population in the VLPO.

\subsubsection*{Wake Inclusive}
The Wake Inclusive system involves neurotransmitters that promote the Wake state as well as those that promote both Wake and REMS. The neuron populations that are active during these states are the GABAergic population in the BF, the cholinergic population in the BF and LDT/PPT, and the monoaminergic populations.

\subsubsection*{Sleep Inclusive}
The Sleep inclusive system involves neurotransmitters associated with NREM specific, REM and NREM specific, and Wake and REM specific states. The neuron populations that are active during these states are GABAergic populations in the VLPO and BF, and cholinergic populations in the BF and LDT/PPT.

\subsubsection*{REM Inclusive}
The REM Inclusive system involves neurotransmitters associated with the Wake and REM specific and REM and NREM specific states. The neuron populations that are active during these states are the GABAergic population in the VLPO, the wake promoting GABAergic population in the BF as well as the cholinergic populations in the BF and LDT/PPT. \\

This analysis allows for an approximation of the coefficients, but it does not yield exact values for these coefficients. Figures \ref{fig:fig12} and \ref{fig:fig13} show the change in concentration levels of Acetylcholine in the BF and GABA in the BF, respectively over a period of 24 hours. The remainder of the neurotransmitters involved show the same fluctuations as that of Acetylcholine, whereas GABA in the sleep active portion of the BF is unique. \\

\begin{SCfigure}[][h]
\centering
\includegraphics[width=2in]{Fig10.pdf}
\caption{Concentration levels of Acetylcholine in the BF over 35 hours.}
\label{fig:fig10}
\end{SCfigure}

\begin{SCfigure}
	\centering
		\includegraphics[width=2in]{Fig11.pdf}
	\caption{Concentration levels of GABA in the BF associated with sleep over 35 hours.}
	\label{fig:fig11}
\end{SCfigure}

\section{Results and Discussion}

The results of the implemented normal, unperturbed, system are shown in Figures \ref{fig:fig12}-\ref{fig:fig15}. Figure \ref{fig:fig12} depicts the slow time variables (which are the easiest to visualize on a large time scale) $AD$ and $GABA_{VLPO}$. The $AD$ curve represents the non-dimensional amount of adenosine accumulation in the BF, which consequently measures sleep pressure. The $GABA$ curve represents the non-dimensional amount of GABA accumulation in the VLPO, which is a measure of wake pressure. Both exhibit a rapid transition between sleep and wake states as well as ongoing periodic behavior for nine days. The system runs like this for much larger periods, demostrating that unless perturbed, the sleep-wake cycle is a self-sustained, periodic oscillation. The period of each cycle is approximately 24 hours - the wake state occupies roughly two-thirds of this time, while the sleep state makes up the remaining one-third. This is consistent with our knowledge of the structure of the human sleep-wake cycle. Figure \ref{fig:fig13} depicts one cycle in the $GABA-AD$ phase plane, showing the exact behavior we required from our previous nullcline analysis. Figure \ref{fig:fig14} is the isolated REM curve for 30 hours. Figure \ref{fig:fig15} shows the $AD$, $GABA$ and REM curves simultaneously in a sample 48 hour time slice. The REM/NREM oscillations respond quickly to the transitions between wake and sleep, and cycle through three to four oscillations in the designated eight hour sleep period. From the biological background and neurotransmitter details, this displays physiologically reasonable results.\\

\begin{figure}[t]
	\begin{center}
		\includegraphics[width=4.5in]{Fig12.pdf}
	\caption{Adenosine and GABA (in the VLPO) levels of the sleep-wake cycle model over 216 hours (9 days).}
	\label{fig:fig12}
 \end{center}
\end{figure}

\begin{figure}[t]
	\begin{center}
		\includegraphics[width=5in]{Fig13.pdf}
	\caption{Phase plane of sleep-wake cycle model for one cycle.}
	\label{fig:fig13}
 \end{center}
\end{figure}

\begin{figure}[t]
	\begin{center}
		\includegraphics[width=5in]{Fig14.pdf}
	\caption{REM cycle visualization of sleep-wake cycle model over a day.}
	\label{fig:fig14}
 \end{center}
\end{figure}

\begin{figure}[t]
	\begin{center}
		\includegraphics[width=5in]{Fig15.pdf}
	\caption{Unperturbed system of AD, GABA and REM from hours 150 to 200 - over two days. }
	\label{fig:fig15}
 \end{center}
\end{figure}

\subsection{Orexin Knockout Experiment}

Narcolepsy is a neurodegenerative disease where there is a loss of neurons containing orexin (OX) \cite{CSF}. It has been shown to exist across many mammals, and has been studied extensively in mice. Here, we take information from the sleep-wake cycles of orexin knock-out (which means the targeted destruction of orexin neurons) mice. These mice exhibit the narcolepsy symptoms of sleep fragmentation, sleep onset during REM sleep states and increased spontaneous REM and NREM sleep during the vigilance state \cite{OXKO,Wu,Baumann,Lu2,Grivel}. \\

A reasonable method to represent the orexin knock-out perturbation to the system is to change the flux of the OX amount equation. Rather than take orexin out all together, we limit the rate at which orexin is produced, but keep the removal factors the same. This is a reasonable way to represent the physiological destruction of orexin neurons, since none of the uptake and removal mechanisms are necessarily effected. It has been shown that levels of orexin decrease in human cerebral spinal fluid by at least 80 percent \cite{CSF}, and so we multiply orexin producing elements of the OX equation ($ACh_{LDT/PPT}$, $NA$, $OX$ and $AD$) by 0.2. Implementing these changes numerically, the periodicity and limit cycle of the system is maintained with orexin decrease. This is illustrated in Figure \ref{fig:fig16}, which clearly shows the same characteristics as seen in Figure  \ref{fig:fig12}.  However, we also don't see the stereotypical changes to the transition rates between behavioral states characteristic to narcolepsy. We have shown that the system can withstand perturbations \cite{OXKO}, but may need to think of implementing another method of making genetically specific perturbations. This also shows that orexin may have to play a larger role in measuring behavioral state instability, and may have to withstand more non-linear changes to its equation when investigated further.

\begin{figure}[h]
	\begin{center}
		\includegraphics[width=5in]{Fig16.pdf}
	\caption{Orexin knockout system over 216 hours (9 days).}
	\label{fig:fig16}
 \end{center}
\end{figure}

\subsection{Sleep Camp Data}
\cite{Carskadon} describes experiments related to perturbations in an individuals sleep/wake cycle.  The trials performed were carried out in a controlled situation with a group of 40 adolescents over a period of one week. The perturbations performed were delayed sleep and wake onset, prolonged sleep and wake periods and forced wake. These perturbations were achieved without the use of external chemical stimuli, and so were essentially achieved in a noninvasive fashion. Salivary cortisol and melatonin levels were recorded daily as well as frequent EEG readings for accurate brainwave activity. The initial time and length of the perturbation period were recorded, and so this data was easy to translate into perturbations in the model. About 50 perturbations were introduced (which translates to a period of about 10 days) and their effect on the system was recorded using graphical and data analysis methods. \\

The perturbations to the system were only introduced to one variable - adenosine. Forced sleep and wake were achieved by forcing $AD$ to its maximum at the appropriate forced waking initialization, and forcing $AD$ to its minimum at the appropriate forced sleep time. These sleep and wake times as well as the prolonged sleep and wake periods were taken from a data set of seven adolescent individuals. Once perturbed the amount of times done so in the physical experiment, we allow the system to run without perturbation for several cycles. This entire set of perturbations and the response is depicted in Figure \ref{fig:fig17}. Figure \ref{fig:fig18} shows what the subject's cycle would look like without perturbation. One can see that the perturbations cause small scale (on the order of a day or two) changes in the length of sleep or wake state. However, once allowed to run unperturbed, the system settles back into a regular 24 hour period. Initial wake and sleep times have drifted, but that is to be expected. When run for a longer time, the system will eventually drift back into its original phase with the same wake and sleep initialization times as the subject started with. We know this is an accurate response to these non-drug related, small perturbations(REF). Figures \ref{fig:fig19} and \ref{fig:fig20} show a quantitative measure of the changes in period in the unperturbed vs. the perturbed cycles. Figure \ref{fig:fig21} demonstrates the upholding of limit cycle dynamics in the face of perturbations, and also that the $AD-GABA$ phase plane retains its size and shape but drifts in phase.

\begin{figure}[!t]
	\begin{center}
		\includegraphics[width=5in]{Fig17.pdf}
	\caption{Perturbed slow time system with forced wake onset and forced sleep onset over a one week period. Extended time to show phase drift and maintained periodicity.}
	\label{fig:fig17}
 \end{center}
\end{figure}

\begin{figure}[t]
	\begin{center}
		\includegraphics[width=5in]{Fig18.pdf}
	\caption{Unperturbed cycle allowed to run for the same amount of time as the perturbed cycle. }
	\label{fig:fig18}
 \end{center}
\end{figure}

\begin{figure}[h!]
	\begin{center}
		\includegraphics[width=5in]{Fig19.pdf}
	\caption{Examination of the length one period in an unperturbed cycle.}
	\label{fig:fig19}
 \end{center}
\end{figure}

\begin{figure}[t]
	\begin{center}
		\includegraphics[width=5in]{Fig20.pdf}
	\caption{Examination of the length of one period in a perturbed cycle. }
	\label{fig:fig20}
 \end{center}

\end{figure}
\begin{figure}[b]
	\begin{center}
		\includegraphics[width=5in]{Fig21.pdf}
	\caption{Perturbed GABA-Adenosine phase plane. }
	\label{fig:fig21}
 \end{center}
\end{figure}
\section{Conclusions}

We have presented a model of the human sleep-wake generating system that identifies specific neuronal regions in the brain with their associated neurotransmitters and projections.  It accounts for the homeostatic influences of sleep-wake, and has the potential to encompass circadian influences as well. It improves upon previous models of sleep-wake cycles in that the oscillations are self-sustaining without the use of external periodic forcing, and it is formulated specially to look at the specific sleep-wake cycle of humans. We use the known neurochemical dynamics of the human sleep-wake cycle to derive equations and form the stability criteria for mathematical phenomena. Since based on limit cycle dynamics, the system runs with the contributions of all the necessary neurotransmitters and with no need of an external driving force. This system gives reasonable predictions for ``natural" sleep bout duration, wake bout duration, REM bout duration and NREM bout duration, as well as estimated natural sleep initiation and wake initiation times. The NREM sleep state is considered the ground state from which all other activity deviates, and REM activity is determined from the difference of the wake and NREM states, as well as the contributions from REM active neurotransmitters. Using experimental data from a mammalian orexin knockout system as well as simply and non-chemically perturbed human sleep-wake cycles, we were able to visualize the perturbed cycle and compare to the experimental results without instability occurring. We can draw a number of conclusions from the resulting model. In the human systems, given forced prolonged waking, sleep duration reacts accordingly to build up appropriate wake pressure. Once the cycle is restored to normal sleep onset and wake onset, the same period for an unperturbed system is maintained. The system exhibits some phase drift with this forcing. Given forced prolonged sleep, wake duration also reacts accordingly to build up appropriate sleep pressure. Once the cycle is restored to normal sleep and wake onset, the periodicity is maintained. The system exhibits some phase drift with this forcing. Given forced wake up during sleep, the subject does not stay awake for long, and falls back into a normal sleep schedule, yielding only some phase drift that is corrected after a few cycles. A subject's REM/NREM cycle goes through three or four oscillations per state (REM or NREM). The first NREM sleep stage is triggered by sleep onset, or when adenosine levels fall below a certain threshold and GABA becomes the dominant neurotransmitter. The last REM cycle is turned off by wake onset, or when adenosine pressure builds back up and GABA decays. Subjects naturally fall asleep and wake up at the intersection points of the GABA and Adenosine thresholds - falling asleep when enough sleep pressure from AD has been accumulated, and waking up when the sleep pressure has decayed once again and when the subject has gone though three to four cycles each of REM and NREM sleep. We can see that given no external forcing (no stimuli, drugs, etc), a ``normal" human's sleep-wake schedule unfolds in a periodic, 24 hour fashion that endures indefinitely unless given some sort of perturbation. \\

It is important to note the implications of the differences between homeostatic and circadian drive, and the importance of both of these drives' effect on sleep. Unlike most recent neurochemically driven models, ours has no need for external forcing to keep it periodically driven. The periodic, limit cycle activity arises naturally from the neurotransmitter dynamics and the equations themselves. This demonstrates an intrinsic, homeostatic force that keeps the human sleep-wake cycle functioning without a change in periodicity, regardless of whether circadian control is included or not. The perturbations in the experimental section demonstrate that even when subjected to strong forcing, the sleep-wake cycle retains its natural 24 hour period, and shifts in phase are the only real artifacts of outside forcing. The next step would be to develop a more complete predictive model of the human sleep wake cycle for use in the scientific and medical fields. \\

\section{Acknowledgements}
This work was supported, in part, though NSF Grant DMS-0636358.  Also, we would like to thank the following for their aid in this project: Dr. Mary Carskadon (Brown University), Dr. Tom Scammell and the Scammell Lab  (BIDMC, Harvard Medical School).

\newpage
\bibliographystyle{plainnat}

redpajama_set_name: RedPajamaArXiv


Tanya and Wouter had their simple and beautiful wedding at Nduna Lodge just outside Alexandria. It was a lovely, festive and relaxed wedding full of love and happiness for Tanya and Wouter. After the ceremony we went for a game drive on the farm and had so much fun taking photos. Thanks to Judith Doubell for second shooting!


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redpajama_set_name: RedPajamaGithub


That's me (left) with my friend, Mike, outside of the Erie Insurance Arena after graduating. Apparently I had not yet learned how to directly show off my diploma.
Five years ago today, I walked across the stage at Erie Insurance Arena and received my diploma from Penn State Behrend.
Graduation is a special time for all, and I can still remember the pride I felt seeing my hard work in college finally reach its culmination. It was an honor to have my family in the crowd to see me graduate, too.
For those graduating this year, you may have been counting down to graduation day for quite some time. Well, that time is just about here, and all of us here at Erie Insurance Arena want to make sure your day goes as smooth as possible.
Parents and friends of graduates need to remember that all patrons entering the Erie Insurance Arena need to go through a metal detector. At every event, this comes as a surprise to some, but it is simply the standard at facilities today.
When going through metal detectors, we ask that you remove your phones, keys, electronic devices (including a camera to take a picture of your smiling grad!), and large metal objects.
However, you don't have to remove your belt, your watch, jewelry, wallet or coins.
We also ask that if at all possible, avoid bringing a bag or purse to graduation. Bags and purses at any event at the Erie Insurance Arena are subject to additional screening, slowing down the entry process. Bring only essential items!
Some tips for you to remember when going through our metal detectors.
One last note: While there are plenty of things you CAN bring, there are some items that you just CANNOT bring. Those include bottles or cans, outside food and beverages, flasks, noise makers, selfie sticks, weapons, and other dangerous or unlawful items. Safety first!
We look forward to having you for graduation at Erie Insurance Arena and we can't wait to be part of your special day!


The Mix and Match Wedding – Can Wedding Dresses Plus Size be in Different Colours?
Wedding dresses plus size or otherwise are the subject of hot debate in the run up to a wedding. Who will wear what? What will the colour scheme be? How will it all be done?
One interesting trend is to specify that bridesmaids and other members of the party don't have to wear dresses of the same colour. Indeed, a very interesting effect can be achieved if the bridesmaids wear a range of colours picking out the highlights of the overall colour scheme, or of the star flower in the wedding bouquet.
It's helpful to look at how a wedding bouquet is made, at this juncture, before we continue to look at the wedding dresses it might inspire.
A wedding bouquet is normally composed of a number of blooms, but with one "star flower" at its heart. Essentially, it may be thought of as a spray of a single flower species, which has been decorated and dressed with complementary species to create an ornate appearance.
The colours of the star flower define the colours of the complementary flowers. A purple calla lily, for instance, which has a bright orange calyx, may dictate that orchids with orange petals and yellow and purple highlights be used to bring the main colours out.
In this example, then, it is possible to specify wedding dresses plus size in not one but three colours – the purple of the star flower; the orange of the supporting flower; and the dusky yellow of the orchid's highlights.
This effect works best with a number of bridesmaids equal to the number of colours in the supporting flowers. It may be enhanced by having each bridesmaid carry in a small bunch of supporting flowers herself; or even a single example of it, depending on the species and the size. Effectively, the whole bridal train then looks like a living embodiment of the bouquet and is used as part of the decoration of the "scene" – that is, when the bride and her entourage appear at the altar or the designated ceremonial spot.
There's another school of thought here – that the bridesmaids can essentially wear whatever dress they like, as long as it is a) smart and b) not white. In this case, the bridal train must be equipped with bouquetsof flowers that tie it in to the overall colour theme of the wedding.
This can actually deliver quite a pleasing effect, as the general tendency is to go for smart, pale outfits that nicely offset the bridal white. Here, the touch of colour lent by the individual bouquets held by the bridesmaids pulls everything together nicely, without an overload of commentary shades causing the whole thing to become a bit top heavy.


Nestled within the "lovely, dark, and deep" forests of the Green Mountains, the Shires boast an unparalleled landscape abundant in adventure, tranquility, and connection.
That connection is of a variety not found through a screen, but rather one found deep in the wilderness, at the apex of a physical challenge, or in the quietude found far away from society — the kind of connection that leaves you "plugged-in" in all of the right ways.
Looking at a map, it's not hard to see why this little-slice-of-heaven lends itself to such ample outdoor opportunities. The region's topography — though peppered with peaks like Glastenbury, Mount Anthony, and Mount Equinox — is wholly green, illustrating the Shires' prime position in the depths of the expansive Green Mountain National Forest.
"The outdoor recreation resources in this region are unimaginably vast," remarked Rob Terry, Executive Director of Merck Forest and Farmland Center (a recreation haven in the Northernmost corner of the Shires). "This region, in part because of the mountain ranges and the great fishing that you find in the Battenkill or the Mettawee, has a real mountain-town feel that's unique in Vermont.
That authentic aesthetic isn't lost on travelers to the Northeast, who have increasingly begun to seek out wild and open spaces to unwind, unplug, and connect. Just a few short hours from major metropolitan areas like New York City, Boston, and Albany, the Shires have gained a reputation for unbeatable skiing, breathtaking hiking, and so much more.
That low population density in itself is a major draw for the region. With no major highway or thruway running through this valley of Vermont, it becomes even easier to take the path less traveled.
The unfractured forests that Terry praises house bountiful rivers, extensive trail networks like the Long Trail and Appalachian Trail, and staggering ski slopes. While endeavors like mountain biking, backcountry skiing, and trail running continue to grow in popularity, adventure of all varieties isn't hard to come by.
BATS, which pursues and advocates the development of multi-use trails in the region, is one of many organizations in the Shires centered around outdoor recreation. NATS, the northern counterpart of Cassano's collective, works closely with the Vermont Mountain Bike Association to develop and maintain trails near Manchester; Bart J. Ruggiere Adaptive Sports Center advocates accessibility for all within the region's outdoor resources; and burgeoning initiatives like Bike Manchester invest in those pastimes that are just beginning to take hold.
"With all of the recreation to be found here, you get to do it in this scenery," said Battenkill Bicycles owner Barrack Evans, who is active in efforts to grow the Shires' cycling scene. While events like the Vermont Challenge have begun to amplify the activity, advocates continue to pursue developments like a rail trail linking Manchester to nearby Dorset—paving the way for a cycling corridor akin to the Long Trail or Catamount Trail.
Such efforts to expand the area's biking, hiking, and skiing trails embody the voracious appetite that Vermonter's have developed for outdoor recreation — more than 70 percent of the state's residents venture into the wild in pursuit of adventure, and 34,000 direct jobs are derived from the industry alongside $2.5 billion in consumer spending.
With such staggering statistics it's no surprise that companies like Orvis (founded and headquartered in the Shires), Darn Tough, and Burton Snowboards have chosen Vermont as their home, and the industry has the capacity to play a major role in the state's economic future.
Even Vermont's government has recognized that potential, with Governor Phil Scott launching the Vermont Outdoor Recreation Economic Collaborative (VOREC) in 2017. Locally, the Shires Outdoor Adventure and Recreation (SOAR) initiative is pursuing a similar goal: promoting the region's vast natural resources while simultaneously making them accessible to everyone, regardless of expertise.
"It seems that we're poised to turn a corner and make outdoor recreation an economic driver in the region," said Anne Houser, owner of the Shires' premiere outdoor gear store The Mountain Goat. An active member of both the recreation and business communities in the area, Houser also recently led the charge for Manchester's designation as an Appalachian Trail Community.
Making information on local trails, events, and resources more accessible to visitors and locals alike is a major goal of both VOREC and SOAR, and many supporters of those initiatives contend that doing so will draw more adventurers into the wild.
"As long as we don't get stuck on one type of recreation, one type of recreator, there's a lot of roads to be had," added Lang.
But where do those paths lead? In a state that is grappling with the issue of youth retention, and actively working to attract new residents to the Green Mountains, there is hope that a bustling and accessible outdoor recreation landscape will not only grow Vermont's economy—but the community behind it as well.
In working to grow and invest in those resources, Vermont's leaders are equally committed to ensuring that the state's trails, rivers, and slopes will remain in-tact for future generations to enjoy.
And Vermont's valleys, mountains, and forests are well worth conserving, promoting, and enjoying. As unplugging from technology becomes increasingly fraught in the annals of modern life, and connection with the natural world grows scarce, there's nowhere quite like the lovely, dark, and deep forests of the Shires to seek out adventure.
Top 8 Quotes from Shirley Jackson's The Lottery (written in North Bennington)!


The RXTE Team is committed to making archival data available promptly and conveniently.
When do data enter the archive?
When will the GO data I'm interested in go public?
How are the data organized in the Archive?
How do I find and retrieve the data I want?
How do I proceed with data reduction and analysis?
When do data enter the archive? / When will the GO data I'm interested in go public?
The RXTE mission completed operations on January 5, 2012. All data collected throughout the mission has since been made public. As no new observations will be carried out, no new data will be entering the archive. Updates to existing data products in the archive (lightcurves, GIF files, etc.) may be made in the future on an "as-needed" basis; any such updates would be to public files and would be immediately public.
Archival RXTE data occupy the xte/data/archive directory on the anonymous FTP server at legacy.gsfc.nasa.gov. Directories at the next lowest level are named AO0 for IOC data, AO1 for AO-1 data, etc.
Below this level, RXTE data files are arranged in a hierarchical set of directories. First come directories named after the proposal number; the next level contains the individual observations ("ObsID's"), while the final level contains directories for each of the Subsystems (PCA, HEXTE, ACS etc.). In the catalogue of publicly available RXTE data, individual entries correspond to the ObsID level. GO data released through our Online Data Area are also grouped by ObsID.
NNNNN is the five-digit proposal number assigned by the GOF and is identical to the name of the parent directory.
TT is the two-digit target number assigned by the GOF. Note that for the case of only one target, the target number may be zero.
different scans for scan-mapping of extended sources.
SS is the two-digit sequence number used for identifying different pointings that make up the same viewing if the SOF decided, for operational reasons, to split that viewing into more than one chunk or if it was split "by accident", by a TOO, for instance.
When not present, it indicates a regular pointed observation (< 8 hours) or the last segment of a long observation.
stdprod - Standard Products - cleaned light curves, spectra etc.
Though they represent the lowest rung in the directory hierarchy, the Subsystem directories do not necessarily contain files of one type. In most cases, a further division is made based on Application, the term used for a distinct source of telemetry. In the case of the PCA, the six Event Analyzers are considered applications. Note that the ASM is not included.
Also in the ObsID directory are various index files used by XDF - the XTE Data Finder - to identify the underlying data files.
Please note that although you can work your way down the directory structure using only FTP, it is often more convenient to use Browse to find and retrieve the data, or retrieve a whole ObsID using anonymous FTP.
The XTEPUBLIC catalogue in the HEASARC database lists all publicly available RXTE data. There two methods for retrieving public data: the web-based Browse, and anonymous-FTP.
Enter search criteria into appropriate boxes. If you chose to search by name, you might enter e.g. 4U1907+09 into the Object Name or coordinates box, and click Submit. To search by parameters, you'll click again to begin your search, and perhaps choose data by proposal number (called PRNB within Browse), ObsID etc. Then click Submit.
A successful search will display the corresponding catalogue entries, i.e. the ObsIDs, one per line. Check the box(es) to the left of the ObsID(s) you're interested in.
(Note that the STDPRODS includes the GIFS in the category above, and the FULL OBSID includes the STDPRODS and GIFS.) Choose one by clicking on it to highlight it.
Then click on the radio button to Retrieve data products in selected categories (or List products in the selected categories and then retrieve the products as a subsequent command).
Click on Submit. Browse will now construct the appropriate tar command, and tar up the data you have requested. This may take a while.
Browse will bring up a new page when your tar job is complete. You may now click on Download TAR file to initiate the transfer of the tar file to your host machine. Make sure you have enough disk space! - the total size of the tar file will be indicated.
ftp legacy.gsfc.nasa.gov, giving 'anonymous' as your name, and your Email address as the password.
get P10154.tar to obtain all the public data from this proposal. The FTP server will automatically create the tarfile on your home machine in the directory from which you initiated the FTP run - make sure you have sufficient space. You might want to cd P10154 and obtain a subset of the data (e.g. get 10154-02-04-00.tar).
There is no advantage to pulling over a gzipped file using e.g. get 10154-02-04-00.tar.gz. All the datafiles within the directory structure are already gzipped.
RXTE datasets may be larger than you are accustomed to, and response may be slow at peak times, particularly if multiple users are attempting to get their hands on the same dataset. Please ensure you have sufficient disk space for the transfers, and try to plan your data retrieval to avoid times of heavy network traffic. If you have problems, try again later.
You may not need all the data files in a given observation: downloading less will speed your job up. Please see our discussion of "Which files can I delete to save space?" in the Getting Started Guide. Also see our Standard Data Products Web page, to find out if the Standard Products for your source will meet your analysis needs. If so, you have the option of just downloading the Standard Products for a given observation, which are a fraction of the size of a whole, raw, dataset.
Please see our Data Analysis & Processing page.
This page is maintained by the RXTE GOF and was last modified on Monday, 03-Jun-2013 15:57:28 EDT.


text:
Journal of Space Exploration
ISSN (Print):2319-9814
ISSN (Online):2319-9822
Current opinion
, Volume: 11( 8) DOI: 10.37532/2319-9822.2022.11 (8).228
Topological investigation of procrystal electron density to improve complicated solid-state ion conductor first-principles simulations
Daniel Jacobsen*
Managing Editor,Journal of Space Exploration, Mexico
*Correspondence:
Angela Guerrero Managing Editor,Journal of Space Exploration, Georgia, E-mail: [email protected]
Received: August 1, 2022, Manuscript No.tsse-22-81717;Editor assigned: August 3, 2022, PreQC No.22-81717 (PQ); Reviewed: August 10, 2022, QC No.tsse-22-81717 (Q); Revised: August 17, 2022, Manuscript No. tsse-22-81717 (R); Published date: August 24, 2022.DOI:10.37532/2319-9822.2022.11 (8).228
Citation::Jacobsen D. Topological Investigation of Procrystal Electron Density to Improve Complicated Solid-State Ion Conductor First-Principles Simulations. J Space Explor.2022; 11(8).228
Designing high-performance solid-state ion conductors for a variety of electrochemical devices requires an atomic-level understanding of ion migration processes. In this endeavor, first-principles simulations are crucial. Although these techniques have limited access to complicated, low-symmetry structures like interfaces, they are typically computationally demanding. Here, we demonstrate how to effectively address this problem using topological analysis of the procrystal electron density. With the help of two instances, we explain how this technique exceeds the state of the art at the moment. We look into Li-ion transport across grain boundaries in Li3ClO electrolyte in the first. Then, we calculate the diffusion coefficients in the spinel LiTiS2 electrode material as a function of charge carrier concentration
Ionic conductors in solid form have several uses in the field of materials research. For instance, they are thought to be crucial in the development of novel functional materials for sensing as well as the rapidly developing energy storage and conversion technologies. For these systems to be successfully applied in high-performance electrochemical devices, ionic transport in those systems must be understood and controlled. Finding ion occupancies, ion migration routes, charge carrier types, point defects, and ion-doping sites at bulk and interfacial structures including grain boundaries and heterophases is often required. A well-established theoretical method for gaining access to these characteristics at the atomistic level is quantum mechanics. With a proven track record, Density Functional Theory (DFT) in particular offers a reasonable trade-off between accuracy and efficiency. To learn more about ion migration processes and the mechanisms that underlie them in complex systems, however, involves laborious studies of complicated potential energy landscapes or the assessment of atomic dynamics, both of which are time-consuming computations. This restricts the use of DFT-based simulations in high-throughput screening campaigns to find novel ionic conductors or to evaluate the performance-related contributions of charge carrier concentration and interfacial structures. Alternatives to DFT-based calculations that need less computing power sometimes sacrifice structural detail, quantitative predictability, and transferability.Finding the lowest energy starting and end state geometries is only one step in the workflows needed to compute kinetic characteristics (such as ion diffusivity), since all conceivable migration pathways must be taken into account and their transition states must also be calculated. Additionally, such procedures should take into account potential metastable sties as well as multistep diffusion mechanisms. Current DFT-based techniques are ideally adapted to carry out such jobs and provide precise predictions in bulk materials that are reasonably simple. Nudged elastic band (NEB) techniques and Ab Initio Molecular Dynamics (AIMD) simulations have shown to be the two most effective strategies in this situation. As recently applied for a set of promising Mg-intercalating materials, one technique to get over the computational restrictions of the NEB method is to estimate the migration barriers using straightforward DFT-derived electrostatic potentials of a framework sub lattice. By examining the electron density distributions from DFT, another recent method enables one to infer an ion migration channel in the crystal structure that is within the predetermined distance limit. However, evaluating the electron density distributions for large structure sets and/or complex structures may take a while. The procrystal density model, on the other hand, is a somewhat sloppy method for total electron density estimates. However, it has been demonstrated elsewhere that due to its speed and moderate accuracy, it may be appropriate for searching for minimal energy routes in a broad variety of structures. The electron density derived using the procrystal technique was in good agreement with the superposition of the free-ion densities for the ionic solid Li2BeF4 ionic crystal. The presence of bond Critical Points (CPs) on interatomic lines in these non-bonded systems was also demonstrated by studies of topological characteristics of procrystal electron density distributions in LiF, MgO, and NaCl crystals. In our most recent research, we looked into the ionic transport characteristics of many well-known and promising potassium-conducting materials that had been chosen using the ICSD geometrical topological screening. The results of the subsequent DFT modelling of the chosen compounds demonstrated that (i) some of the pathways among the selected sets had high migration energies and, as a result, made no contributions to the migration maps, and (ii) a number of effects that cannot be considered within the geometrical-topological approach (different vacancy formation energies, close position of other ions with respect to the pathway, etc.) influence the ion migration properties.
The proposed method can be viewed as a promising tool for finding new ion conducting materials among the large (due to low computational complexity) crystal structure sets accompanied by accurate DFT calculations because the NEB migration energies and RCP densities are consistent. In this instance, using the predicted gradient paths for each independent transition can both speed up the calculations and prevent errors brought on by the initial guess that was made without consideration of the ion trajectory
Journal of Space Exploration received 214 citations as per Google Scholar report

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Tim Heidecker Shares New Song "To The Men," with All Proceeds to Benefit Alabama's Yellowhammer Fund
By Harry Todd | May 16, 2019 | 4:13pm
Images via Juliana Giraffe, Orion Read Music News Tim Heidecker
In light of the recent, vicious anti-choice laws set in place by Alabama and Georgia, among other states, comedian and musician Tim Heidecker has released a new song, "To The Men."
There are no jokes in this song, though. There shouldn't be any jokes in this song. "To The Men" tells the story of a young woman, living in an indiscriminate state that somehow feels entitled to control her body, as she comes to realize she is dealing with an unwanted pregnancy. From behind an acoustic guitar, Heidecker sings:
Told a friend who had been down that road
Told a friend she was late
And the friend said she had no choice
Have the baby or get out of the state
It's straightforward songwriting that effectively highlights the predicament countless women find themselves in when male lawmakers, like those pictured in the song's art, feel compelled to wrest control from women's bodies. Later, the song angrily addresses those male politicians:
I just have a simple question
To the men without flaws
Have you seen a young girl dying
Have you seen 'em bleed to death
Have you seen 'em take their final breath
All proceeds from the song directly benefit Alabama's Yellowhammer Fund, an organization that "provides funding for anyone seeking care at one of Alabama's three abortion clinics and will help with other barriers to access (travel, lodging, etc.) as well as able," per their website.
"To The Men" will not be on Heidecker's forthcoming album What The Brokenhearted Do, but all sales of that album will also go to charity.
You can donate to the Yellowhammer Fund here and find an excellent list of what you can do to help women in states with extreme abortion laws here.
Listen to "To The Men" below—you can download the track via Bandcamp with a donation of $1 or more—and confront the lawmakers behind the recent abortion bans through the official single art further down.
Hi. I have a new one about the new insane abortion laws. it's very sad. https://t.co/v5J8TCQu8t
— Tim Heidecker (@timheidecker) May 16, 2019
tim heidecker
Also from Tim Heidecker
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Tim Heidecker, Weyes Blood Serve up Folksy "Let It Be" Cover By Christine Fernando June 21, 2019 | 3:16pm
Tim Heidecker Shares New Song "To The Men," with All Proceeds to Benefit Alabama's Yellowhammer Fund By Harry Todd May 16, 2019 | 4:13pm
Tim Heidecker Announces New Album What The Brokenhearted Do..., Nods at Nilsson on "When I Get Up" By Scott Russell April 23, 2019 | 10:00am
Us By Dom Sinacola March 21, 2019 | 10:40pm
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Hamilton family gives $250,000 to University of Denver
Gift strengthens ties between the Denver Art Museum and School of Art and Art History
DENVER – Frederic C. and Jane Hamilton have donated $250,000 to the University of Denver (DU) in order to strengthen the link between DU's School of Art and Art History (SAAH) and the Denver Art Museum (DAM). The program will advance students' understanding of how artists and museums work together to present important international installations to the public. The gift will fund a visiting artist program for five years, two artists per year. Selected artists will prepare an installation at the DAM and also will participate at DU through class projects, guest lectures, demonstrations and workshops.
"We are fortunate to have these two important institutions working to grow the visual arts in our community," said Mr. Hamilton. "It is our hope that this donation further engages students and the public in the creative process of artists working today."
"The University of Denver has a long-standing relationship with the Denver Art Museum – we have closely aligned educational missions," says M.E. Warlick, chair of SAAH. "This is very enriching for our students who want to meet professional artists."
The first Hamilton Visiting Artist is Steina, a video artist, whose work will be featured at the DAM as part of "Blink! Light, Sound and the Moving Image." The museum's first large-scale exhibition devoted entirely to electronic and time-based media, "Blink!" opens March 13. Steina's work has been exhibited extensively both nationally and internationally for more than four decades, and the artist will be in residence and installing her work at DAM between Feb. 14-28. The native-born Icelander now resides in Santa Fe. She will work with students in classes, do critiques and present a performance at DU.
In April, DU and DAM will welcome renowned ceramic artist Walter McConnell, professor and chair of the Division of Ceramic Art at New York State College of Ceramics at Alfred University in Alfred, NY. McConnell is known for his unfired, apparitional installations of moist clay housed in plastic enclosures. The artist will be in residence April 4-16. His work will be featured in the DAM exhibition, "Overthrown: Clay Without Limits."
Earlier contributions from the Hamiltons to DU funded the Hamilton Gymnasium in the Ritchie Center and the Hamilton Family Recital Hall in the Robert and Judi Newman Center for the Performing Arts. Frederic Hamilton is Chair of the Board at the DAM and Jane Hamilton is a member of DU's Board of Trustees.
The University of Denver is committed to improving the human condition and engaging students and faculty in tackling the major issues of our day. DU ranks among the top 100 national universities in the U.S. For additional information, go to www.du.edu/newsroom.
Contact: Kristal Griffith
Email: Kristal.Griffith@du.edu
Tags: news-release